University:
East Georgia State College
Course:
MATH 1540 | Calculus I
Academic year:

2023
Views:

53

Pages:

14
Author:

Giselle A.

Sec 9.3: Measures of Regression & Prediction Intervals Three Types of Variation About a Regression Line 1. 2. 3. TOTAL variation EXPLAINED variation UNEXPLAINED variation ***Without a regression line, the best predictor for y given a value for x is y (the mean of the y-values) 1 Variation About a Regression Line TOTAL Deviation = yi  y EXPLAINED Deviation= yˆi  y UNEXPLAINED Deviation= y Can be explained by relationship between x & y yi  yˆi Cannot by rel. between x & y …chance or other variables (xi, yi) Unexplained deviation yi  yˆi Total deviation yi  y y (xi, ŷi) Explained deviation yˆi  y (xi, yi) x x 2 TOTAL variation = EXPLAINED + UNEXPLAINED Variation About a Regression Line TOTAL variation: sum of the squares of the differences between the y-value of each ordered pair and the mean of y.   yi  y  2 EXPLAINED variation: sum of the squares of the differences between each predicted y-value and the mean of y.   yˆi  y  2 UNEXPLAINED variation: sum of the squares of the differences between the y-value of each ordered pair & each corresponding predicted y-value.   yi  yˆi TOTAL variation = EXPLAINED + UNEXPLAINED 2 Coefficient of Determination: The ratio of the explained variation to the total variation. Denoted by r2 Explained variation r  Total variation 2 5 Ex: Coefficient of Determination The correlation coefficient for the Old Faithful data was r ≈ 0.979 Find the coefficient of determination. What does this mean? About 95.8% of the variation in times is EXPLAINED by the variation in durations…relationship between x & y! About 4.2% of the variation is UNEXPLAINED …due to other factors or to sampling error. 6 Standard Error of Estimate: The standard deviation of the observed yi -values about the predicted ŷ-value for a given xi -value. Denoted by se ( yi  yˆi) se  n2 2 n is the number of ordered pairs in the data set  The closer the observed y-values are to the predicted y-values, the smaller the standard error of estimate is 7 The Standard Error of Estimate In Words In Symbols 1. Make a table that includes the column heading shown. 2. Use the regression equation to calculate the predicted y-values. 3. Calculate the sum of the squares of the differences between each observed y-value and the corresponding predicted y-value. xi, yi, yˆi, ( yi  yˆi ), ( yi  yˆi )2 yˆ  mxi  b ( yi  yˆi )2 ( yi  yˆi)2 se  n2 4. Find the standard error of estimate. 8 The regression equation for the advertising expenses and company sales data as calculated in section 9.2 is: ŷ = 50.729x + 104.061 Find the standard error of estimate. You will be doing this by hand on the quiz and using the computer for the project. x 2.4 1.6 2.0 2.6 1.4 1.6 2.0 2.2 y 225 184 220 240 180 184 186 215 ŷi 225.81 185.23 205.52 235.96 175.08 185.23 205.52 215.66 (yi – ŷ i)2 (225 – 225.81)2 = 0.6561 (184 – 185.23)2 = 1.5129 (220 – 205.52)2 = 209.6704 (240 – 235.96)2 = 16.3216 (180 – 175.08)2 = 24.2064 (184 – 185.23)2 = 1.5129 (186 – 205.52)2 = 381.0304 (215 – 215.66)2 = 0.4356 Σ = 635.3463 UNEXPLAINED VARIATION 9 n = 8, Σ(yi – ŷ i)2 = 635.3463 ( yi  yˆi) 2 se   n2 635.3463  10.290 82 The standard error of estimate of the company sales for a specific advertising expense is about $10,290 10 Ex# 2 Team Detroit concludes that there is a significant relationship between the amount of radio advertising time (in minutes per week) and the weekly sales of a product (in hundreds of dollars.) A) Find the regression equation B) Use the regression equation to find the predicted ŷ -values C) Calculate the sum of the squared differences of each observed y-value and the corresponding predicted y-value D) Calculate Se E) INTERPRET the results Radio ad time Weekly sales 15 26 20 32 20 38 30 56 40 54 45 78 50 80 60 88 Linear regression equation: ŷ = 1.4054x + 7.3108 x 15 20 20 30 40 45 50 60 y 26 32 38 56 54 78 80 88 ŷi 28.39 35.41 35.41 49.46 63.51 70.54 77.56 91.61 (yi – ŷ i)2 Linear regression equation: ŷ = 1.4054x + 7.3108 x 15 20 20 30 40 45 50 60 y 26 32 38 56 54 78 80 88 ŷi 28.39 35.41 35.41 49.46 63.51 70.54 77.56 91.61 (yi – ŷ i)2 (26 – 28.39)2 = 5.7121 (32 – 35.41)2 = 11.6281 (38 – 35.41)2 = 6.7081 (56 – 49.46)2 = 42.7716 (54 – 63.51)2 = 90.4401 (78 – 70.54)2 = 55.6516 (80 – 77.56)2 = 5.9536 (88 – 91.61)2 = 13.0321 Σ = 231.8973 UNEXPLAINED variation n = 8, Σ(yi – ŷ i)2 = 231.8973 ( yi  yˆi) 2 se   n2 6.217 in hundreds of dollars The standard error of estimate of the weekly sales for a specific radio ad time is about $621.70.

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